Conjugation of reddening sequences and conjugation difference

TL;DR

This paper explores the conjugation of reddening sequences in cluster algebras, extending key lemmas and properties, and introduces a conjugation difference to quantify red mutations in mutated initial seeds.

Contribution

It introduces a conjugation of reddening sequences based on $c$-vectors and extends several fundamental lemmas and properties to totally sign-skew-symmetric cluster algebras.

Findings

Extended the Rotation Lemma and Target before Source Theorem

Established mutation invariance of reddening sequences

Constructed conjugation difference to count red mutations

Abstract

We describe the conjugation of the reddening sequence according to the formula of $c$-vectors with respect to changing the initial seed. As applications, we extend the Rotation Lemma, the Target before Source Theorem, and the mutation invariant property of the existence of reddening sequences to totally sign-skew-symmetric cluster algebras. Furthermore, this also leads to the construction of conjugation difference which characterizes the number of red mutations a maximal green sequence should admit in any matrix pattern with the initial seed changed via mutations.